Кудрявцев А.А. Пособие по теории вероятностей
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A
B
G
g
g
G
1/4
[0, 1]
Ω
Ω
G
Ω
X Y
12 13
20
X Y
x
y
1
3
1
1
3
1
0
G
g
G
X Y x y
12
1 G
G = {(x, y)|x, y ∈ [0, 1]}
|x − y| ≤ 1/3 x, y ∈ [0, 1] g
g G 5/9
2a
2l < 2a
x
ϕ
G {(x, ϕ)|x ∈ [0, a], ϕ ∈ [0, π]}
x ≤ l sin ϕ
G g
p
p =
mes g
mes G
=
1
aπ
π
Z
0
l sin ϕ dϕ =
2l
aπ
.
2a
2l
ϕ
x
x ≤ l sin ϕ
g
ϕ
x
π
a
G
0
2a
2a S
n W W
1
, . . . , W
n
w
1
, . . . , w
n
2l
1
, . . . , 2l
n
A
W
W
W
A
i
i = 1, . . . , 5
A
i
A P(A) =
P
5
i=1
P(A
i
)
A
1
A
3
A
4
w
j
W
k
j k
x = l
j
sin ϕ
(x, ϕ)
G
A
1
A
3
A
4
P(A
2
) = 0 A
2
(x, ϕ) x = 0
A
5
A
ij
i, j = 1, . . . , n
w
i
w
j
P(A
ii
) = 0
i
A
5
= (A
12
+ . . . + A
1n
) + (A
23
+ . . . + A
2n
) + . . . + A
n−1,n
. (2.2)
P(A
5
) = (P(A
12
) + . . . + P(A
1n
)) + (P(A
23
) + . . . + P(A
2n
)) + . . . + P(A
n−1,n
) =
=
1
2
[(P(A
11
) + . . . + P(A
1n
)) + . . . + (P(A
n1
) + . . . + P(A
nn
))] .
P(A
i1
) + . . . + P(A
in
)
w
i
2l
i
/(aπ)
P(A) = P(A
5
) =
1
2
n
X
i=1
2l
i
aπ
=
S
2aπ
.
P(A) n
n
P(A) S/(2aπ)
[0, 1]
[−1, 1]
p q
x
2
+ px + q = 0
2/9
A B C
ABC
AB a
BC b
A
C
[0, 1]
ξ η
P(ξ < x) P(η < x) x
A
A
x x
A
A
x x
A
A
x x
A
A
x x
1 2
A A
x x
1 2
A A
x x
1 2
A A
x x
A = { } B = {
} A
A
B
p B
A p = 1/3
p =
1
3
=
P(AB)
P(B)
,
A B
n A k
B l AB m
B l
A l
m AB
p m/l
p =
m
l
=
m/n
l/n
=
P(AB)
P(B)
.
A B
P(B) > 0 A
B
P(A|B) =
P(AB)
P(B)
.
A B
A = { } B = {
} A
B A
B
P(A|B) = P(A)
P(B) > 0 P(AB) = P(A)P(B)
A B
P(AB) = P(A)P(B). (3.1)
P(A) > 0 P(B) > 0
A B A
B
A B
P(AB) > 0
A B
P(B) > 0 A B
P(A|B) = P(A)
A B A B
A B
1
A B
2
B
1
B
2
= ∅ A B
1
∪ B
2
A = {
} B = { }
P(AB) = P({ }) =
1
4
=
1
2
·
1
2
= P(A)P(B).
Ω = {(ω
1
, ω
2
)|ω
1
, ω
2
∈ [0, 1]}.
A = {(ω
1
, ω
2
)|0 ≤ ω
1
≤ 1, 0 < a ≤ ω
2
≤ 1},
B = {(ω
1
, ω
2
)|0 < b ≤ ω
1
≤ 1, 0 ≤ ω
2
≤ 1},
C = {(ω
1
, ω
2
)|ω
2
≤ (ω
1
− b)/(1 − b), ω
1
, ω
2
∈ [0, 1]}
A B A C
ω
1
ω
2
1
1
b
a
0
P(AB) = (1−a)(1−b) =
P(A)P(B) A B
P(AC) = (1 −b)(1 −a)
2
/2 6= (1 −a) ·(1 −b)/2 =
P(A)P(C)
A C
B
1
, . . . , B
n
1 ≤ i
1
< . . . < i
r
≤ n r = 2, . . . , n
P
r
\
k=1
B
i
k
!
=
r
Y
k=1
P(B
i
k
). (3.2)
r = 2
= {
} = {
} = {
}
P( ) = 1/4 = 1/2 · 1/2 = P( )P( ).
P( ) =
1
4
6=
1
8
= P(
)P( )P( ),
A
BC
1
1
1
2
1
2
0
Ω = {(ω
1
, ω
2
)|
0 ≤ ω
1
, ω
2
≤ 1} A B C
P(A
1
. . . A
n
) = P(A
1
) · P(A
2
|A
1
) · . . . · P(A
n
|A
1
. . . A
n−1
), (3.3)
P(A
1
) · P(A
2
|A
1
) · . . . · P(A
n
|A
1
. . . A
n−1
) =
= P(A
1
) ·
P(A
1
A
2
)
P(A
1
)
·
P(A
1
A
2
A
3
)
P(A
1
A
2
)
· . . . ·
P(A
1
. . . A
n−1
A
n
)
P(A
1
. . . A
n−1
)
=
= P(A
1
. . . A
n
).
3 2
p
k
k = 1, . . . , 4
k
C
i
i B
k
= {
k } p
k
= P(B
k
) B
k
C
i
C
i
B
1
= C
1
, B
2
= C
1
C
2
, B
3
= C
1
C
2
C
3
, B
4
= C
1
C
2
C
3
C
4
.
P(B
1
) = P(C
1
), P(B
2
) = P(C
1
)P(C
2
|C
1
),
P(B
3
) = P(C
1
)P(C
2
|C
1
)P(C
3
|C
1
C
2
),
P(B
4
) = P(
C
1
)P(C
2
|C
1
)P(C
3
|C
1
C
2
)P(C
4
|C
1
C
2
C
3
).
P(C
1
) =
2
5
, P(
C
1
) =
3
5
, P(
C
i+1
|C
1
. . . C
i
) =
3 − i
5 − i
,
P(C
i+1
|
C
1
. . . C
i
) =
2
5 − i
, i = 1, 2, 3.
p
1
= 0.4 p
2
= 0.3 p
3
= 0.2 p
4
= 0.1
0.3
0.4 0.6
p
A
k
= {k
} k = 1, 2, 3 p = P(A
1
A
2
A
3
) A
i
p = P(A
1
)P(A
2
)P(A
3
) = (1 − 0.3) · (1 − 0.4) · (1 − 0.6) = 0.168.
p = P(A
1
A
2
A
3
|A
1
) = P(A
2
)P(A
3
) = 0.24.